Решите неравенство _x ( x^2 + (3)/(2) ) 4 _(x^2 + (3)/(2))(x) .

Положим t = _x ( x^2 + (3)/(2) ) . Тогда, поскольку t (4)/(t) ((t-2)(t+2))/(t) 0 [ arrayl 0 < t 2 t -2 array . [ arrayl (1)/(t) (1)/(2) t -2 array ., получаем _x ( x^2 + (3)/(2) ) 4 _(x^2 + (3)/(2))(x) [ arrayl _(x^2 + (3)/(2))(x) (1)/(2) _x ( x^2 + (3)/(2) ) -2 array . cases x > 0 [ arrayl _(x^2 + (3)/(2))(x^2) 1 _(x^(-2))( x^2 + (3)/(2) ) 1 array . cases cases x > 0 [ arrayl (x^2 - ( x^2 + (3)/(2) ))/(( x^2 + (3)/(2) ) - 1) 0 (( x^2 + (3)/(2) ) - x^(-2))/(x^(-2) - 1) 0 array . cases cases x > 0 [ arrayl x in (x^4 + (3)/(2)x^2 - 1)/(x^2 - 1) 0 array . cases cases x > 0 ((x^2 + 2)( x^2 - (1)/(2) ))/(x^2 - 1) 0 cases cases x > 0 (1)/(2) x^2 < 1 cases (1)/(sqrt(2)) x < 1.

x \in \left[ \frac{1}{\sqrt{2}}, 1 \right)